Simplify the following expression and state the condition under which the simplification is valid: $y = \dfrac{q^2 + 11q + 30}{q^2 + 4q - 12}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{q^2 + 11q + 30}{q^2 + 4q - 12} = \dfrac{(q + 5)(q + 6)}{(q - 2)(q + 6)} $ Notice that the term $(q + 6)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(q + 6)$ gives: $y = \dfrac{q + 5}{q - 2}$ Since we divided by $(q + 6)$, $q \neq -6$. $y = \dfrac{q + 5}{q - 2}; \space q \neq -6$